XIII.-EQUAL, GREATER, LESS.

Magnitudes are equal to each other when the one may be made exactly to coincide with the other, or when they may be divided into a number of parts, so that the sum of all the parts of the one may be made to coincide with the sum of all the parts of the other.

But when the magnitudes are such that the whole of the one may be made to coincide with a part of the other, or when the sum of all the parts of the one is contained within the sum of all the parts of the other, leaving some part of the latter unoccupied, the containing magnitude is the greater, the contained the less.

XIV.-ANGLE.

Two straight lines meeting in a point are said to make an angle at that point, the magnitude of which is measured by the quantity of plane surface abutting on the point of intersection between the straight lines or arms of the angle.

XV.-RIGHT ANGLE.

When a straight line standing on another straight line makes the adjacent angles equal to each other, each of the angles is called a right angle, and the straight line which stands on the other is called perpendicular.

PROPOSITIONS.

I.

Straight lines from the same point and in the same direction coincide.

Let A B, A C, be two straight lines lying in the same direction from the point A. In either of these as A B take any point b, and let c be a point in A C, such that Ac is equal to Ab. Then the tracks, by which the position of the points b and c are determined, are identical in respect of distance and direction, and the points b and c coincide; that is to say, any point in either of the lines A B, A C (and, therefore, every point in each of those lines) corresponds with a corresponding point in the other. Therefore, the lines A B, A C, wholly coincide.

II.

If two straight lines coincide in any two points, they coincide throughout to the extent of their joint length.

Let the straight lines 1 and 2 coincide in the points A and B, and let C be any point in line 1, on the same side of A with the point B; C'a point in line 2, on the same side of A with B, and at the same distance as C from A.

Then, because A, B, and C, are points in the same straight line, B and C lie in the same direction from A; and, for a like reason, the points B and C' are in the same direction from A. Therefore the points C and C', being in the same direction from A, with a common point B, are in the same direction from that point with each other; and they are, by the construction, at the same distance from A, therefore the points C and C' coincide. That is to say, any point in either of the lines 1 and 2 (as far as they jointly extend in distance) coincides with a corresponding point in the other, and therefore the lines coincide throughout.

III.

Straight lines joining any two points in a plane surface fall wholly within the plane.

Let A B be a straight line joining any two points in a plane. Then the spectator, moving along the plane from A to B, will be without motion in the direction of the normal (Def. 9); or in other words, the distance of B from A in the direction of the normal is null. Therefore, a spectator moving direct from A to B, along the line AB, will be without motion in the direction of the normal; and AB, by the definition, will be a line in the plane.

IV.

Straight lines in opposite directions from the same point form parts of a single straight line.

Let A B, A C, be straight lines in opposite directions from the point A, and let the spectator proceed along AB from A to B. Then BA is manifestly the direction in which he must return, in order to reach the point A from whence he started; that is to say, BA is the opposite to AB, and is therefore in the same direction with AC, and B A C forms a single straight line.

V.

All right angles are equal.

Let BAC, EDF, be any two right angles (fig. 2), and let BA and ED be produced to G and H respectively.

Let the plane EDFH be superimposed upon the plane B A C G, so that the point A shall coincide with the point D, and the straight line EDH with BAG. Then the plane surface abutting on the point D, between D E and DH, on the side towards F, consisting of the angular spaces EDH, FDH, will coincide with the surface abutting on the point A, between A B and AG, on the side towards C, consisting of the angular spaces BA C, CAG. Therefore the angles EDF, FDH, are together equal to the angles BAC, CAG. But

the two together, E DF, FDH, are together double of E DF; and BAC, CA G, are double of BAC. Therefore B A C is equal to ED F.

Cor. 1.-As division produces no alteration in the magnitude of a figure, the angular expansion round a point in a straight line on one side of the line is equal to two right angles, in whatever manner it may be divided, whether into two equal or unequal angles, or into any number of angles.

Cor. 2.-The whole angular expansion round a point in a plane is equal to four right angles.

VI.

If a straight line standing on two other straight lines make the adjacent angles equal to two right angles, the two straight lines form a single straight line.

Let BD (fig. 3) be a straight line, making with the straight lines BA, BE, the angles DB A, DBE, equal to two right angles; the straight lines BA, BE, will form a single straight line ABE.

Let A B be produced on the other side of B to any point F. Then the angular expansion A B D, DBF, is equal to two right angles, and therefore, by the hypothesis, to the two angles DBA, DBE. Take away the angle A B D, and the remaining angle D BF will be equal to the remaining angle